A $p$-adic ($p\equiv 3(\mathrm{mod}4)$) depth-$5$ supercongruence for Gaussian $p$-th power sums over a square

Abstract

For an odd prime $p$, define $G_n(p)={\sum }_{a=1}^{p-1}{\sum }_{b=1}^{p-1}(a+bi{)}^n\in \mathbb{Z}[i]$. We study the $p$-adic valuation of these Gaussian power sums for $n\le p,n=r(p-1)$ and show that it is governed by the interaction of the fourfold symmetry of the square, ordinary power-sum congruences, and Bernoulli numbers. If $p\equiv 3(\mathrm{mod}4)$ and $p\ge 7$, then we prove an unexpectedly deep supercongruence $G_p(p)\equiv -\frac{p^5}{12}(p-1{)}^2(p-2)(1-i)B_{p-3}(\mathrm{mod}{p}^6)$.